Uniform Estimate of the Relative Free Energy by the Dissipation Rate for Finite Volume Discretized Reaction-Diffusion Systems

  • André Fiebach
  • Annegret Glitzky
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 77)


We prove a uniform Poincaré-like estimate of the relative free energy by the dissipation rate for implicit Euler, finite volume discretized reaction-diffusion systems. This result is proven indirectly and ensures the exponential decay of the relative free energy with a unified decay rate for admissible finite volume meshes.


Admissible finite volume mesh reaction-diffusion system free energy discrete Poincaré and Sobolev-Poincaré inequality 



The work was partially supported by the European Commission within the 7th Framework Programme MD\(^3\) “Material Development for Double Exposure and Double Patterning” and by the DFG Research Center Matheon Mathematics for Key Technologies within project D22 “Modeling of Electronic Properties of Interfaces in Solar Cells”.


  1. 1.
    Bessemoulin-Chatard, M., Chainais-Hillairet, C., Filbet, F.: On discrete functional inequalities for some finite volume schemes. preprint arXiv:1202.4860v2 (January 15, 2014)
  2. 2.
    Eymard, R., Gallouët, T., Herbin, R.: The finite volume method. In: Ciarlet, P., Lions, J.L. (eds.) Handbook of Numerical Analysis VII, pp. 723–1020. Elsevier (2000)Google Scholar
  3. 3.
    Fiebach, A., Glitzky, A., Linke, A.: Uniform global bounds for solutions of an implicit Voronoi finite volume method for reaction-diffusion problems. Published online in Numer. Math. (2014). doi: 10.1007/s00211-014-0604-6
  4. 4.
    Glitzky, A.: Uniform exponential decay of the free energy for Voronoi finite volume discretized reaction-diffusion systems. Math. Nachr. 284, 2159–2174 (2011)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Glitzky, A., Griepentrog, J.A.: Discrete Sobolev-Poincaré inequalities for Voronoi finite volume approximations. SIAM J. Numer. Anal. 48, 372–391 (2010)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Glitzky, A., Gröger, K., Hünlich, R.: Free energy and dissipation rate for reaction-diffusion processes of electrically charged species. Appl. Anal. 60, 201–217 (1996)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Physikalisch-Technische BundesanstaltBerlinGermany
  2. 2.Weierstrass InstituteBerlinGermany

Personalised recommendations